Chapter 2 Review of Forces and Moments - Brown University

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design of airplane wings); or because the object is spinning (this effect is exploited by people who throw, kick, or hit balls for a living). Two effects contribute to drag: (1) Friction between the object’s surface and the fluid or air. The friction force depends on the object’s shape and size; on the speed of the flow; and on the viscosity of the fluid, which is a measure of the shear resistance of the fluid. Air has a low viscosity; ketchup has a high viscosity. Viscosity is often given the symbol η , and has the rather strange units in the SI 2 system of Nsm − . In `American’ units viscosity has units of `Poise’ (or sometimes 2 centipoises – that’s 10 − 2 Poise). The conversion factor is 1P= 0.1Nsm − . (Just to be confusing, there’s another measure of viscosity, called kinematic viscosity, or specific viscosity, which is ς = η/ ρ , where ρ is the mass density of the material. In this course we’ll avoid using kinematic viscosity, but you should be aware that it exists!) Typical −5 −2 numbers are: Air: η ≈ 1.73× 10 Nsm for a standard atmosphere (see http://users.wpi.edu/~ierardi/PDF/air_nu_plot.PDF for a more accurate number) ; Water, −2 −2 −2 η ≈ 0.001Nsm ; SAE40 motor oil η ≈ 0.5Nsm , ketchup η ≈ 60Nsm (It’s hard to give a value for the viscosity of ketchup, because it’s thixotropic. See if you can find out what this cool word means – it’s a handy thing to bring up if you work in a fast food restaurant.) (2) Pressure acting on the objects surface. The pressure arises because the air accelerates as it flows around the object. The pressure acting on the front of the object is usually bigger than the pressure behind it, so there’s a resultant drag force. The pressure drag force depends on the objects shape and size, the speed of the flow, and the fluid’s mass density ρ . Lift forces defy a simple explanation, despite the efforts of various authors to provide one. If you want to watch a fight, ask two airplane pilots to discuss the origin of lift in your presence. (Of course, you may not actually know two airplane pilots. If this is the case, and you still want to watch a fight, you could try http://www.wwe.com/, or go to a British soccer match). Lift is caused by a difference in pressure acting at the top and bottom of the object, but there’s no simple way to explain the origin of this pressure difference. A correct explanation of the origin of lift forces can be found at http://www.grc.nasa.gov/WWW/K-12/airplane/right2.html (this site has some neat Java applets that calculate pressure and flow past airfoils). Unfortunately there are thousands more books and websites with incorrect explanations of lift, but you can find those for yourself (check out the explanation from the FAA!) Lift and drag forces are usually quantified by defining a coefficient of lift C L and a coefficient of drag C D for the object, and then using the formulas 1 2 1 2 FL = ⎛ ⎜ ρV ⎞ ⎟CLAL FD = ⎛ ⎜ ρV ⎞ ⎟CDAD ⎝2 ⎠ ⎝2 ⎠ Here, ρ is the air or fluid density, V is the speed of the fluid, and A L and A D are measures of the area of the object. Various measures of area are used in practice – when you look up values for drag coefficients you have to check what’s been used. The object’s total surface area could be used. Vehicle
manufacturers usually use the projected frontal area (equal to car height x car width for practical purposes) when reporting drag coefficient. C and C are dimensionless, so they have no units. L The drag and lift coefficients are not constant, but depend on a number of factors, including: 1. The shape of the object 2. The object’s orientation relative to the flow (aerodynamicists refer to this as the `angle of attack’) 3. The fluid’s viscosity η , mass density ρ , flow speed V and the object’s size. Size can be quantified by A L or A D ; other numbers are often used too. For example, to quantify the drag force acting on a sphere we use its diameter D. Dimensional analysis shows that C D and C L can only depend on these factors through a dimensionless constant known as `Reynold’s number’, defined as ρV A Re = η D For example, the graph on the right shows the variation of drag coefficient with Reynolds number for a smooth sphere, with diameter D. The projected area A 2 D = π D /4 was used to define the drag coefficient Many engineering structures and vehicles operate with Reynolds numbers in the range 3 6 10 − 10 , where drag coefficients are fairly constant (of order 0.01 - 0.5 or so). Lift coefficients for most airfoils are of order 1 or 2, but can be raised as high as 10 by special techniques such as blowing air over the wing) 100 10 C D 1 0.1 0.1 1 10 100 1000 10 4 10 5 Re = ρ VD / η The variation of drag coefficient with Reynolds number Re for a smooth sphere. Lift and drag coefficients can be calculated approximately (you can buy software to do this for you, e.g. at http://www.hanleyinnovations.com/walite.html . Another useful resource is www.desktopaero.com/appliedaero ). They usually have to be measured to get really accurate numbers. Tables of approximate values for lift and drag coefficients can be found at http://aerodyn.org/Resources/database.html Lift and drag forces are of great interest to aircraft designers. Lift and drag forces on an airfoil are computed using the usual formula ⎛1 2 ⎞ FL = ⎜ ρV ⎟ CLAW ⎝2 ⎠ ⎛1 2 ⎞ FD = ⎜ ρV ⎟ CDAW ⎝2 ⎠ V α F L c F D
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Chapter 2 Review of Forces and Moments - Brown University

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